I think one could make the argument that this chapter is the most pivotal in the entire Tolkien corpus. Bilbo finds the ring! The rest is just either leading up to this moment or resolving what problems this moment brings about.
Regarding the gaming inspriation from this chapter, the most influential part for me was always the riddle game. I remember trying to guess the answers to the riddles that Bilbo and Gollum posed to one another and feeling very smart when my 6th grade self got them right. Riddles always found their way into my games and I loved it when, as a player, I had a riddle to wrestle with. Within the chapter, the riddle game becomes the principal means of communication between Gollum and Bilbo; they both understand it's sacredness. I also think it's interesting that Bilbo effectively cheats to end the game. "What have I got in my pocket?" isn't a riddle; both Bilbo and Gollum know it. I wonder if one can't read the fact that Bilbo cheats at a sacred contest as an early malign influence of the ring. Of course, that may be simply externalizing Bilbo's own, well, lesser traits.
A second thing that this chapter brings to light is Hobbit Luck. It's mentioned before, but hasn't really come into play very much until now. Bilbo is at his wits end, stumped by the "time" riddle, and blurts out the answer by chance. He answers the fish riddle only because a fish jumps and lands on his hairy feet. We see Bilbo getting saved again and again by luck, to the point where I wonder if it isn't some supernatural, almost quntifiable force.
Considering Tolkien's writing, one of the things that jumped out at me here was the fact that we have an unreliable narrator, one that actively contradicts himself (or, at least, witholds relevant, interesting, and potentially useful information from the reader). Consider when we first meet Gollum. The narrator tells us "I don't know where he came from, nor who or what he was." Yet we get more facts about Gollum and his history even in this chapter! As the riddle game begins, we learn that such a game was the only one Gollum played "with other funny creatures sitting in their holes long, long ago before he lost all his friends and was driven away." Gollum answers the "eye in a blue face" riddle by remembering when he lived in a hole on a riverbank with his grandmother, who we later learn sucked eggs. I think it's an interesting stylistic choice for Tolkien to adopt a narrator's voice that feigns ignorance yet clearly knows The Whole Story. It's almost Gandalf-like. Though there are other things (like the mentions of The Red Book of Westmarch) that suggest Bilbo himself is the narrator.
- In Fellowship, when Gandalf chastises Frodo for not having any sympathy for Gollum, he mentions it was pity that stayed Bilbo's hand. We see that, almost verbatim, in this chapter. Bilbo could have stabbed Gollum, but imagines what it's like to live as Gollum does, and leaps over him instead.
- As Risus Monkey mentioned earlier, orcs and goblins seem to be used to refer to the same creature or, at best, different kinds of the same creature.
- I had forgotten that the first edition of The Hobbit has a different version of this chapter, one that is explained away in the forward to Lord of the Rings. An interesting comparison of the two versions can be found here.